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Theorem eqsb3 2871
Description: Substitution applied to an atomic wff (class version of equsb3 2524). (Contributed by Rodolfo Medina, 28-Apr-2010.)
Assertion
Ref Expression
eqsb3 ([𝑥 / 𝑦]𝑦 = 𝐴𝑥 = 𝐴)
Distinct variable group:   𝑦,𝐴
Allowed substitution hint:   𝐴(𝑥)

Proof of Theorem eqsb3
Dummy variable 𝑤 is distinct from all other variables.
StepHypRef Expression
1 eqsb3lem 2870 . . 3 ([𝑤 / 𝑦]𝑦 = 𝐴𝑤 = 𝐴)
21sbbii 2068 . 2 ([𝑥 / 𝑤][𝑤 / 𝑦]𝑦 = 𝐴 ↔ [𝑥 / 𝑤]𝑤 = 𝐴)
3 nfv 2009 . . 3 𝑤 𝑦 = 𝐴
43sbco2 2506 . 2 ([𝑥 / 𝑤][𝑤 / 𝑦]𝑦 = 𝐴 ↔ [𝑥 / 𝑦]𝑦 = 𝐴)
5 eqsb3lem 2870 . 2 ([𝑥 / 𝑤]𝑤 = 𝐴𝑥 = 𝐴)
62, 4, 53bitr3i 292 1 ([𝑥 / 𝑦]𝑦 = 𝐴𝑥 = 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wb 197   = wceq 1652  [wsb 2061
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2069  ax-7 2105  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2062  df-cleq 2758
This theorem is referenced by:  pm13.183  3498  eqsbc3  3636
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